Assuming a isn't 0,
The smallest 3 digit mutltiple of 7, abc would be 105.
A neat thing about multiples of 9 is that if you add all the numbers i.e. a + b + c + d =9x, you get another multiple of 9.
Using abc = 105
a + b + c = 1 + 0 + 5 = 6
The closest multiple of 9 to 6 is 9, thus d = 3
Therefore, abcd = 1053, would be the smallest possible number that fits.

Please check, thanks

Is this only apply on multiply of x9 ? Or all these kind of question ? Can I used this method for all this kind of question? Is it i just need to get 105 = a + b + c = 1 + 0 + 5 = 6. Last part I'm not very sure too. What do you mean by the closest multiply of 9 to 6 is 9 . Thus d = 3. Thank for helping

Yes, the 9 rule can be applied to all multiples of 9. It only works for 9s. As for how 9 being the closest multiple of 9 to 6, I should have said the next multiple of 9.
Lets try another example, lets say abcd is a multiple of 9, a =9, b=,8,c=7, find d. 987d

Using the 9 rules we'll understand that a + b + c + d = 9x.
So 9 + 8 + 7 + d =9x
24 + d = 9x
So what is the next multiple of 9, that fulfils 24 + d = 9x, given that d is positive.
Multiples of 9 = 9, 18, 27, 36
It would be 27, hence d = 3, thus 9873 would be a multiple of 9

Oh i see. So this is only apply to only multiply of 9 . So as long next time we encounter another similar question. We can use this method to solve the question easy. You are marvellous. Your students are so lucky yo have you. !!! Will show this to my girl, hope she will understand